Jul 31, 2006 - arXiv:cond-mat/0608005v1 [cond-mat.str-el] 31 Jul 2006. Finite temperature properties of the triangular lattice t-J model, applications to.
Finite temperature properties of the triangular lattice t-J model, applications to Nax CoO2 Jan O. Haerter, Michael R. Peterson, B. Sriram Shastry
arXiv:cond-mat/0608005v1 [cond-mat.str-el] 31 Jul 2006
Physics Department, University of California, Santa Cruz, Ca 95064 (Dated: February 6, 2008) We present a finite temperature (T ) study of the t-J model on the two-dimensional triangular lattice for the negative hopping t, as relevant for the electron-doped Nax CoO2 (NCO). To understand several aspects of this system, we study the T -dependent chemical potential, specific heat, magnetic susceptibility, and the dynamic Hall-coefficient across the entire doping range. We show systematically, how this simplest model for strongly correlated electrons describes a crossover as function of doping (x) from a Pauli-like weakly spin-correlated metal close to the band-limit (density n = 2) to the Curie-Weiss metallic phase (1.5 < n < 1.75) with pronounced anti-ferromagnetic (AFM) correlations at low temperatures and Curie-Weiss type behavior in the high-temperature regime. Upon further reduction of the doping, a new energy scale, dominated by spin-interactions (J) emerges (apparent both in specific heat and susceptibility) and we identify an effective interaction Jef f (x), valid across the entire doping range. This is distinct from Anderson’s formula1 , as we choose here t < 0, hence the opposite sign of the usual Nagaoka-ferromagnetic situation. This expression includes the subtle effect of weak kinetic AFM2 - as encountered in the infinitely correlated situation (U = ∞). By explicit computation of the Kubo-formulae, we address the question of ∗ 3 practical relevance of the high-frequency expression for the Hall coefficient RH . We hope to clarify some open questions concerning the applicability of the t-J model to real experimental situations through this study.
I.
INTRODUCTION
Since the discovery of superconductivity in H2 Ointercalated NCO crystals4 this compound has drawn significant attention. In addition, NCO displays a variety of remarkable thermoelectric effects, such as an unusually large and magnetic field dependent thermopower5 and a T -linear Hall-coefficient3,6 . While some of its properties are similar to those seen in high-Tc materials there are some distinctive differences. Both the cobaltates and the cuprates become superconducting in the vicinity of the Mott-insulating state, however, while this is achieved through hole-doping in the cuprates, NCO requires electron-doping x between x = 1/4 and x = 1/37. While for both compounds the role of electron transport in two-dimensional planes - separated by thick insulating layers - appears to be crucial, the sites for the localized electron wavefunctions are organized in planar square lattices (Cu-ions) in the cuprates and in a triangular lattice (Co-atoms) in NCO. The most striking differences however, are the sign of the hopping, which is inferred to be positive in the cuprates and negative in NCO, and its absolute value, which is believed to be an order of magnitude smaller in NCO8 . While the experimental results indicate a rich blend of phases - stemming perhaps from structural rearrangements with x - it is first necessary to systematically probe the scope of a single model for strongly correlated systems - namely the t-J model - in the case of this electronically frustrated triangular lattice. In the half-filled band, where strong interactions lead to the spin- 12 Heisenberg-antiferromagnet (HAF), the square geometry facilitates anti-ferromagnetism (AFM) through its bipartite nature, while the tendency towards
AFM in the cobaltate system is weaker. However, extensive numerical computations9 indicate N´eel long-ranged order even in the triangular lattice. Away from halffilling, projected hopping on the square lattice system (t > 0) obeys the Nagaoka mechanism10 . This results in a competition between the ferromagnetic (FM) kinetic operator and AFM potential energy for finite values of J. Conversely, in the current case of the triangular lattice with a negative hopping t < 0, AFM tendencies are supported - rather than weakened - by the electronically frustrated kinetic energy operator. By further reducing the t-J model - taking J = 0 - we explicitly isolate the effect of electronic frustration which leads to kinetic Antiferromagnetism (kA)2 , an effect triggered only by the motion of holes in an AFM spin-background, even away from the special case of doping 1/L, with L the number of sites. This paper has two main objectives: Firstly, we want to supplement the arguments made in ref. 11, where the Curie-Weiss phase in the doping range x > 0.5 was addressed. Secondly, we will discuss the finite temperature t-J model starting from the Nagaoka-regime and then extend into finite dopings. The structure is as follows: In section II we describe our model and the exact diagonalization technique, which we utilize for our computations. Section III presents the results for the static properties chemical potential, specific heat and entropy as well as the spin susceptibility. We then investigate the dynamic Hall-coefficient RH (ω) and compare its dcand high-frequency behavior. Lastly, we draw a connection between the Hall-coefficient and the diamagnetic susceptibility. For all properties, we study systematically the dependence on temperature, doping, and interaction strength. In Section IV we conclude and summarize our
2 results.
Beginning at a doping of only a single hole in the infinitely correlated system, we track the effect of kA into the intermediate doping regime. The specific heat develops a low-temperature peak indicative of effective spininteractions, generated by a dressed hole in an AFM-spin background. The same effect is manifest in the susceptibility results where we present an x- and J-dependent effective interaction, with a small but finite AFM value in the case of J = 0. We proceed by an explicit computation of the frequency-dependent Hall-coefficient. The results show weak frequency-dependence of RH and encourage ∗ the applicability of its high-frequency limit RH to practical situations. While small in magnitude compared to the spin-susceptibility, we show through an identity in terms of derivatives - suggested by our numerical results - that the diamagnetic susceptibility alone may be closely connected to the temperature dependent Hall-coefficient.
II.
MODEL AND EXACT DIAGONALIZATION
As a model for hole doped strongly interacting Mottinsulators, the t-J Hamiltonian can be applied to the experimental situation of electron-doped NCO after an electron-hole transformation12, making use of the symmetry of the Hubbard model in respect to the Mottinsulator. The replacement rules are t → −t, δ = |1 − n| = x and qe → −qe , where δ is now hole-doping away from half-filling and n is the electron density per site. This leads to the t-J model X X n ˆin ˆj ˆ = −t Sˆi · Sˆj − PˆG cˆ†iσ cˆjσ PˆG + J H 4 ,σ
(1) where cˆ†iσ (ˆ ciσ ) creates (annihilates) an electron of spin σ, Sˆi is the three-component spin-operator, n ˆ i is the number operator and i specifies the lattice site. PˆG denotes the Gutzwiller projector and the summation is over all nearest neighbor pairs < i, j >. In the following, whenever doping x is mentioned this refers to electron doping in the NCO system and holedoping in the model Hamiltonian. To compute the complete spectrum - as required for thermodynamic properties - we employ the canonical ensemble exact diagonalization technique for small clusters. The computational demand limits the analysis to quite small systems and finite size effects are challenging to overcome. Nonetheless, careful comparison between several systems of different geometry enables us to extract the stable behavior and somewhat isolate finite size effects. Conversely, finite size effects can help identify the favored eigenstate of a system. We have employed several toroidal and ladder geometries with L = 9, 12, and 18 sites, as well as the highly
symmetric icosahedral geometry. The icosahedron - a 12sited Platonic solid with 20 equilateral triangular faces shares the property of geometric frustration with the infinite system while the effect of the reduced coordination number (z=5) appears to have only a small, quantitative effect on the computational results (Fig. 1(c)). Our 12site torus clusters are depicted in Fig. 1(a)-(b). A third 12-site cluster is the simple 4 × 3-cluster (called Torus c, not pictured). While the numerical effort of exact computations for translationally invariant systems can be significantly reduced by exploiting space group symmetries of the Hamiltonian, these symmetries are broken upon the introduction of a magnetic field, as relevant for the evaluation of the Hall coefficient. A magnetic field is introduced by the usual Peierls substitution which modifies the hopping t between sites i and j by Z 2π j t → tij (A) = t exp i A · ds , (2) φ0 i where A is the magnetic vector potential and φ0 the flux quantum. We define H the flux threading a triangular plaquette as α ≡ 2π φ0 △ A · ds. In finite systems, the value of the smallest non-zero magnetic field is limited to values of α ≥ π/l where l is the length of a periodic loop in the system. Through a particular gauge we can achieve l equal to the number of triangular faces in the cluster, this guarantees the equality of the flux values through all plaquettes. An example is given in Fig. 1(a). A similar strategy has been followed in the case of the square lattice quantum Hall effect13 . We complement the computations on periodic systems with ladder systems, which enable an infinitesimal flux to be chosen due to open boundary conditions in one of the spatial dimensions (Fig. 1(b)) and the Landau-gauge is adequate. In the case of the icosahedron (Fig. 1(c)) the magnetic field is applied through a (fictitious) magnetic monopole in its center, creating equal magnetic fluxes through all triangular plaquettes. III. A.
RESULTS
Chemical Potential
The chemical potential is evaluated as a derivative of the free energy F with respect to the particle number N , µ(T ) =
∂F (T ) . ∂N
The derivative has to be taken as a discrete difference of F (N ). To take this derivative numerically, we use the central difference formula. Hence it is suitable to define µ+ = F (N + 1) − F (N ) and µ− = F (N ) − F (N − 1) . The value of µ can then be obtained from µ=
(µ+ + µ− ) . 2
3
5
4
a T2
2
3 0
12 7
6
0
11 8
9
0
10 10
3
5
4
0
9
11
12
13
14
0
0
7
8 16
15 0
6
17
18
19
2
0
5 20
0
4
21
22
23
y
1
0
3 0
0
2
1 0
0
1
T1
x
12 site torus 0
0.5
1
1.5
2
2.5
3
3.5
FIG. 1: (color on-line) Small numbers along arrows (red) denote Peierls phases in units of the flux through a single plaquette, phases on Icosahedron (c) not shown for clarity. Dotted parallelograms (black) specify boundaries of finite clusters ((a) and (b)). All sites on corners of parallelograms are equivalent and translations along bounding vectors T1 and T2 tile infinite lattice; shaded region in (b) forbidden for ladder. Note: different coordinate systems are used for torus and ladder. For ladder-system we use the Landau gauge with A ∝ yˆ x, (a) represents only one choice (denoted Torus a) for 12 sites. Another (denoted Torus b) results from ladder (b) by connecting all corners through T1 and T2 (not shown). Torus c is simple 3 × 4 cluster (not shown).
In Fig. 2(a)-(b) we present µ(T ) at several J and dopings x. At low temperatures µ(T ) shows degenerate Fermi behavior (∝ T 2 )14 . In the strong-correlation regime (J = 0 and J = 0.4|t|) the ground state (g.s.) internal energy has a maximum at intermediate dopings x ≈ 0.5 due to the competition of increasing number of mobile particles and reduction of the available vacancies upon decrease of doping and, hence, the low temperature chemical potential changes sign here. In the high-temperature regime the doping-dependence of the Hilbert-space dimension determines the limit of µ(T → ∞) ∝ T
∂g(x) ∂x
with g(x) = (1 − x) log(2x/(1 − x)) + log(1/x) essentially the natural logarithm of the dimension of the Hilbert-space. In the projected t-J model, g(x) peaks at x = 1/3. For x < 1/3 the slope at large T becomes negative. In Fig. 2 the comparison of µ(T ) for different interactions shows that the impact of projected fermions near half-filling is to force a negative slope of µ for x > 0 which occurs in the non-interacting system only for x < 0. This is an example of the qualitative difference between a strongly-interacting Fermi-system and
its non-interacting counterpart.
At larger x > 0.7 the effect of strong interactions is gradually lost and the temperature-dependence resembles that of the non-interacting Fermi-system (Fig. 2(c)). For low temperatures well below the linear range of µ, finite J > 0 generally increases the value of the chemical potential with a stronger increase for x near the Mottinsulator. The high-temperature behavior is independent of J. In Fig. 2(d) we compare the results with direct experimental measurements of the T -dependent chemical potential15 at x ≈ 0.83. Since our results in this doping range are rather insensitive to the value of U , we compare with our U = 0 results. Here, µ(T ) was obtained directly from the tight-binding R ∞ density of states by the implicit equation hni = 2 × −∞ f (ǫ, µ)ρ(ǫ)dǫ with fixed hni = 1 + x, where f denotes the Fermi-Dirac distribution function and ρ is the non-interacting density of states. This fit confirms the notion of NCO as a system with an unusually small Fermi temperature scale. In Fig. 2(d) we show as a shaded region the values t = −100
8 6 4 2 0 -2 -4
(a)
(b)
x 0.83
(c)
25
(d)
x 0.83
x 0.83
20 t=-100 K
15 10 5
0.16 0.16 0
1 2 T/|t|
0.16 0
1 2 T/|t|
-300 K
0
1 2 T/|t|
0
∆µ [meV]
µ/|t|
4
0
-5 100 200 300 T [K]
FIG. 2: (color on-line) Chemical potential µ(T ) for all accessible dopings on Torus b: x = 2/12 = 0.16 to 10/12 = 0.83 for J = 0 (a), J = 0.4|t| (b) and U = 0 (c). Data for U = 0 computed for infinite non-interacting system, comparison with (a)-(c) shows relatively weak effect of interactions in the doping range x > 0.8.(d) Squares are experimental data for ∆µ(T ) = µ(T ) − µ(0) from ref. 15. Shaded area is a comparison with our data for U = 0 for t = −100 K to t = −300 K.
K to t = −300 K, photoemission results8 point to a value of |t| ≈ 10meV ∼ −118 K, hence |t| here is more than an order of magnitude smaller than in the cupratesuperconductors. In the following, when making contact with experiments, we take t = −100 K for simplicity. B.
Specific Heat, Entropy
In Fig. 3(a)-(b) we present the specific heat per site Cv (T ) =
1 dhEi L dT
with hEi the thermodynamic expectation value of the internal energy. We first concentrate on the case of J = 0, corresponding to an infinite on-site interaction. Near the band insulator (x ≈ 1) the temperature-dependence resembles that of non-interacting fermions, computed at fixed particle density by first determining µ(T ) and then the T -derivative of the internal energy. When the doping is reduced, the main energy scale - manifest in a peak of Cv (T ) at temperature Tm (x, J) - continuously moves to smaller temperatures, away from the tight-binding peak. Even at our lowest doping x = 1/L the peak-feature persists (called kA-peak). This feature is magnified in Fig. 4 for several different systems.
Numerical work on the g.s. of the problem of a single hole in an AFM spin-background2 suggests this system to exhibit a weak effective AFM interaction Jef f ∝ 1/L due to the projected kinetic energy term only. While for one hole this excitation decays in the thermodynamic
limit, Jef f nonetheless induces a N´eel ordered AFM g.s. and low energy excitations similar to those observed in the HAF9 . The validity of our computations for the case of a single hole is underscored by the entropy. In Fig. 5, the temperature dependence of the entropy shows that the lowenergy features observed in the specific heat are necessary for the entropy to obtain its (known) high-temperature limit. For T /|t| ≈ 0.075 half the entropy is recovered (s(T )/s(∞) = 0.5), indicative of entropy-enhancement due to electronic frustration16 . For intermediate dopings, the entropy reaches half its maximum at T /|t| ≈ 0.75 and for dopings x ≈ 0.7 - relevant to the Curie-Weiss phase of NCO - we find T /|t| ≈ 0.78. Within a high-temperature expansion16 a numerical argument was given for an S-shaped low temperature behavior of the entropy for the case of x = 0.2 and x = 0.4. Our computations at small dopings rather suggest s ∼ T for low T as applicable to a metal from Fermi-liquid arguments. It is remarkable here that Cv (T ) reflects these low energy excitations, notably even for larger dopings than the one applicable to the Nagaoka-situation. We now distinguish the effect of J > 0 (Fig. 3(b)). This scenario has been investigated by several authors for the case of the hole doped square lattice and the HAF17,18 . For the situation of a single hole J > 0 shifts the kA-peak to larger temperatures Tm = J/t. In Fig. 8(b) we present the peak positions as function of doping and J. We find Tm (x, J) to increase monotonically both with x and J and the slope as function of J for fixed x to decrease for larger values of x. Extrapolating these results at a given value of J to x = 0 we obtain roughly
x band-limit
5
J=.4|t|
J=0 0.2
0.2
x=.83
x=.83 0 0.2
0 0.2
x=.77
0
0
0.2 0 0.4 0.2 0 0.4 0.2 0 0.4 0.2 0 0.4 0.2 0 0.4 0.2 0
0.2 0 0.4 0.2 0 0.4 0.2 0 0.4 0.2 0 0.4 0.2 0 0.4 0.2 0
x=.72 x=.5
Cv/kB Mott-limit
x=.77
x=.42 x=.25 x=.17 x=.08 0
1
(a)
2
3
x=.72 x=.5 x=.42 x=.25 x=.17 x=.08 0
(b)
T/|t|
1
2
3
T/|t|
FIG. 3: (color on-line) Specific heat per site as function of T for varying x (computed on 18 site (x ≥ 0.72) and 12 site cluster (x < 0.72), (a) J = 0, (b) J = 0.4|t|. Dotted lines show Cv (T ) for non-interacting fermions and solid line for exact results. Note: scale changes in plots of different dopings, long temperature exponential behavior at large x due to finite-system induced gap in energy spectrum. Results for x = 0.72 appeared in ref. 11
0.5 0.4 Cv/kB
Tm (x = 0) ≈ J. Our result should be distinguished from investigations on the t > 0 square lattice17 , where Tm was found to follow the opposite trend decreasing slightly with x at small dopings, along with a rapid suppression of the peak. This supports the notion of electronic frustration contributing to AFM correlations in the present study in opposition to competing AFM exchange coupling and the FM Nagaoka-mechanism in the case of t > 0, capable of quickly eliminating the low-energy spin excitations.
1/18, Torus 1/14, Torus 1/12, Torus a 1/12, Torus b 1/12, Torus c 1/13, Torus 1/9, Torus 1/12, Icos.
0.3 0.2 0.1 0 0.1
0.2
0.3
0.4
0.5
T/|t| FIG. 4: (color on-line) Cv (T ) at x = 1/L for several sets of boundary conditions, peak in specific heat is systemindependent property.
An experimentally interesting quantity related to the specific heat is γ defined by Cv (T ) ∼ γT or γ = ds/dT corresponding to the low energy electronic part of the specific heat. When extracting this quantity from our numerical results, we neglect the contributions at very low T arising purely from the finite system induced gap in the spectrum (scaling as exp(−∆/T ) where ∆ is the gap between the g.s. and the first excited state).
6 externally applied magnetic field.
1.0
χ−1 =
0.852 0.8
d2 G dM 2
can then be evaluated numerically, by constraining the canonical ensemble to the sector of minimal |Sztot |, and those sectors where Sztot is raised (lowered) by ~.
0.718 0.631 0.6 s/kB
x=.42 x=.08 0.4
x=.72
0.2
0.0 0
0.5
1
1.5
2 2.5 T/|t|
3
3.5
4
FIG. 5: (color on-line) Entropy per site for three different dopings at J = 0 (x = 1/12 = 0.08, x = 5/12 = 0.42, x = 13/18 = 0.72), horizontal lines indicate high-temperature limit s(T → ∞) = kB log(g(x))/N , x = 0.08 corresponds to Fig. 4. Exponential increase at low T for intermediate x due to finite system-size gap ∆.
For the large doping case x = 0.7 we found11 γ ≈ 45 (±5) mJ/molK 2 (γ0 ≈ 33 mJ/molK 2 for noninteracting tight-binding electrons), indicating a significant many-body enhancement. Experimental results19 indicate γ ≈ 26 mJ/molK 2 . On the other hand, in the low-doping regime the linear increase with T is only observed in a very small temperature range and this range is possibly better classified as a non-Fermi-liquid regime. Roughly, γ(x)/γ0 (x) ≫ 1 for x & 0. The increase in γ with decreased doping is also demonstrated clearly by the low T slope of s(T ) in Fig. 5.
C.
Spin Susceptibility
To gain further insight into the role of spin-fluctuations and to address the existence of the experimentally observed Curie-Weiss phase of NCO20 , we compute the spin susceptibility χ(T ) as function of temperature from the P Gibbs free energy G(T, M ), here M = i Siz is the total magnetization. This quantity follows from the Helmholtz free energy F (T, H) through a Legendre-transformation ∂F = F (T, H) + M H. H is the G(T, M ) = F (T, M ) − H ∂H
For low temperatures, near the band-limit (x = 1) we identify weakly temperature-dependent, Pauli-like behavior of χ, reminiscent of the experimentally observed spin density wave (SDW)-phase (Fig. 6). When x is lowered past x = 0.75 we find11 a crossover to a Curie-Weiss like behavior with appreciable renormalizations of the g.s. susceptibility compared to its bare value, mirroring the experimental findings. This behavior can roughly be understood by picturing electrons distributed on an empty lattice: Initially, the many-body eigenstate enables the electrons to move around without interacting directly with their neighbors, yielding a susceptibility that is Jdependent only to second-order. When the density is increased (n > 0.25 or equivalently n < 1.75), electrons are forced to interact through the exchange-coupling directly and hence yield a strongly J-dependent magnetic susceptibility. The emergence of the Curie-Weiss phase is further supported by the strong magnetic field suppression of the Heikes-Mott term in the thermopower within the t-J model, an effect due to spin-entropy5 as was demonstrated in ref. 11. For temperatures larger than a certain crossover temperature (which we call T ∗ ), for all dopings, χ(T ) can be described through the Curie-Weiss form χ(T ) =
1 L µ2B p2ef f 3 V kB (T − θ)
with the Weiss-temperature θ and the effective magnetic moment pef f , V is the volume of the crystal and L the number of sites. For all accessible system sizes - in particular Torus a,b,c (Fig. 1) - we compute χ(T ) to extract θ, pef f and T ∗ , the parameters characterizing the magnetic properties at a given doping and interaction strength. To determine θ and pef f we fit the hightemperature part of our computations for χ(T ) to the function f (T ) = a/(T − b) (compare Fig. 7(a)). In Fig. 7(a)-(c) we present the inverse susceptibility at several different dopings near the Mott-insulating phase. The results for θ and pef f are shown in Fig. 7(d)-(f). For all x and J ≥ 0, θ remains negative indicating AFM correlations. θ shows a linear increase when the doping is increased or the interaction is decreased. In short, as mentioned in ref. 11, θ can be collapsed into the form θ(x, J) = −c Jef f (x) where Jef f (x) = J(1 + c′′ x|t|) + c′ x|t|
2 χ[µB /v]
7
1.0 0.5 0.0 1.0 0.5 0.0 1.0 0.5 0.0 1.0 0.5 0.0 1.0 0.5 0.0
x=0.83
x=0.42
x=0.75
x=0.33
x=0.67
x=0.25
x=0.58
x=0.16 J
x=0.5
0
1
2 T/|t|
0
1
x=0.08
2 T/|t|
FIG. 6: (color on-line) Spin susceptibility per site χ(T ) for all accessible dopings x on 12-site torus. Dependence on J/|t| =(0, 0.1, 0.2, 0.3, 0.4, 0.5), indicated by arrow, blue dotted line shows bare-susceptibility for non-interacting fermions of density n = 1 + x and black dotted line indicates susceptibility of isolated spin- 21 fermions of density n = 1. Plots for x ≥ 0.58 appeared in ref. 11.
with c = 4.0, c′ = 0.01425 and c′′ = −0.9175. In contrast to the relation given by Anderson et al.1 in the case of the t > 0 square lattice, in our situation of t < 0 even at J = 0 there is a slightly negative intercept suggesting AFM spin-interactions in the low temperature regime of the electronically frustrated U = ∞ Hubbard model, indicative of kinetic Antiferromagnetism (kA). On the other hand, once J > 0 is allowed, this intercept grows strongly and expresses dominant AFM correlations (Fig. 7(e)). We compare these results with experiments for various dopings (x > 1/2) of NCO. Among others, the cases of x = 0.7519,23,24 , x = 0.722 , x = 0.6 and x = 0.925 , various x in the three-layer system21 and x = 0.5526 were experimentally investigated. The comparison with experiment proves difficult due to the large number of structural phases21 , the role played by the Na-ions in the crystal and the large quantitative and qualitative difference between powder and single crystal samples23 . However, the Weiss-temperature appears to lie between 100 K and 200 K near x = 0.7 (varying considerably with H||c and H||a23 ) and was measured to be smaller at larger dopings21 . Hence, at least for x > 0.5,
the t-J model description yields reasonable experimental agreement. The effective magnetic moment pef f in units of µB has a value close to that obtained for localized spin- 21 p √ particles at half-filling µ1/2 = g × s(s + 1) = 3 and decays linearly for x → 1. The inverse susceptibility shows a minimum at a temperature T ∗ which becomes more pronounced with increasing J. At low temperatures the system prefers antiferromagnetic order which is strengthened by J > 0. Thermal spin fluctuations break this order and when T is increased to values of order J a small perturbative B-field is sufficient to align spins antiparallel to the field. Hence, the susceptibility is largest at this temperature. When T is further increased, the ability of the field to flip spins is weakened by thermal fluctuations, consequently, χ(T ) decreases. The behavior of this peak as function of hole doping away from the Mott-insulator has already been investigated within a high-temperature expansion16 . Due to the small size of the systems available to our study, it is difficult to confirm the results for the doping dependence16 . Here, we study the behavior as function
8
15
J/|t| .5
J/|t|
J/|t| .5
[f]
1.5
.5
5 0
χ-1[v/µB2]
x=0.25 *
.1 J/|t|
15
.4
.3 .2
T .5
5
.3 .2
J/|t| .4 .5 .4 .3 .2 .1
.2 x=0.16
.1
x=0.25
.5
.2 .1
0
-0.5
.1 .2 [a]
.3
x=0.16
0
.3
-1
[b]
.4
(e)
[c] [d]
.5
-1.5
[e]
x=0.16
J/|t|
-2
T
0.2
0.4
x
0.6
0.8
1
0
.5 .4
-0.01
.3 x=0.08
.2 .1
-0.005
x=0.08
T*
.5 .4 .3 .2 .1
θ/|t|
.5 .4 .3 .2 .1
J/|t|
J/|t|
15
5
(d) [a]
0
J/|t|
10
[b]
1
0.5
*
0
[e] [d] [c]
.1
.1
J/|t|
.3
10
x=0.25
peff/µB
.4
.3 .2
θW/|t|
10
.4
.4
x=0.08
T/|t|
(b)
T/|t|
(c)
(f)
-0.02 -0.025 -0.03
0 -0.5 0 0.5 1 1.5 2 2.5 3-0.5 0 0.5 1 1.5 2 2.5 3-0.5 0 0.5 1 1.5 2 2.5 3
(a)
-0.015
J=0
-0.035 0 0.1 0.2 0.3 0.4 0.5 0.6
x
T/|t|
FIG. 7: (color on-line) (a)-(c) Inverse susceptibility vs. temperature for dopings x = 1/12 = 0.08 (bottom), x = 2/12 = 0.16 (center), and x = 3/12 = 0.25 (top) for Torus a, b and c (l to r). Solid vertical arrows show increase in J, dotted (blue) line shows example for fit to extract Weiss-temperature θ(x, J) and effective moment pef f . Dotted vertical arrows indicate temperature T ∗ ; Note: comparison of the three systems allows isolation of finite size effects, most visible for x = 0.16. (d) Effective magnetic moment pef f . (e) Weiss-temperature θ vs doping x for J = 0 . . . 0.5. Letters in angled brackets correspond to experimental data from [a] Viciu21 et al., [b] Gavilano22 et al., [c] Sales23 et al., [d] Takeuchi24 et al., [e] Motohashi19 et al., and [f] Effective moment of isolated spin- 21 -particles on half-filled lattice. (f) Magnification of θ vs. x for J = 0 for several clusters.
of J, considering several dopings near half-filling and several system geometries (Fig. 8(a)). The figure indicates that T ∗ /|t| roughly scales as J/|t|, hence, the scale at which the system undergoes a transition from an AFMordered to a paramagnetic state is set by the interaction strength. As was shown for the case of J = 0 and t > 0 in the hole doped system2 , a single hole favors AFM N´eel order and numerical results9 indicate that the triangular lattice Heisenberg model also favors N´eel order - at least in the g.s. and the elementary excitations - hence it is not surprising that a finite J > 0 emphasizes this tendency. This is in opposition to the case of t < 0 where the kinetic energy term favors ferromagnetism and competes with the Heisenberg-interaction preferring AFM.
D.
Hall Coefficient
Experimentally, NCO has been shown to exhibit unusual thermoelectric properties, resulting from strong electron correlations. Wang et al.6 measured the dc-Hall coefficient as function of temperature at several dopings showing unexpected temperature dependence with an oscillatory behavior below 200 K and a rapid linear increase for temperatures above 200 K. Much theoretical work has been invested into the study of RH on square lattices3,12,27,28,29,30,31,32,33,34,35,36,37,38,39 both for g.s. properties and for finite temperatures. For the triangular lattice, it was possible to predict the high-temperature behavior within a high-temperature expansion of the t-J-model3,12 in its high-frequency limit. Several authors have also attempted to describe this behavior through Boltzmann transport theory36,37 within the relaxation time approximation, where RH is
9 0.8
0.08 (a) 0.16 (a) 0.25 (a) 0.08 (b) 0.25 (b) 0.08 (c) 0.16 (c) 0.25 (c)
0.7 0.6 0.5 T*/|t|
(a)
0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
J/|t|
and the relaxation time τk is taken to be isotropic. For large dopings - x ≥ 0.7 - where the hole-like Fermi-surface of NCO approximates a circle, this leads to the same high-temperature linear coefficient as the t-J-high-temperature expansion, however, the behavior completely diverges in the dense regime (x < 0.7), possibly due to the misleading replacement of the interacting Fermi-surface by its non-interacting counterpart. Also, at low temperatures - where the relaxation time is anisotropic - the oscillatory behavior seen in experiments is not reproduced. To shed some light on the low-temperature and strongly-correlated behavior, we start from the highfrequency limit3 for RH (T, ω):
0.8 0.7
∗ RH (T ) ≡ lim RH (T, ω) ω→∞ ivL < [Jx , Jy ] > = lim − 2 B→0 Be < τx,x >2
0.6
(b)
Tm/|t|
0.5 0.4 0.3 0.08 0.16 0.25 0.33
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
J/|t|
FIG. 8: (color on-line) (a) AFM-paramagnetic crossovertemperature as function of J for several dopings on 12-site torus clusters a, b, c. (b) Evolution of Tm as function of J, values averaged over three 12-site clusters a, b, c.
expressed as an integral over the Fermi-surface curvature
σα,β
ie2 = ωv
< τα,β
Here, e is the electron charge, Jx (Jy ) are the currents in x (y)-directions, v is the unit cell volume, and τx,x is the diagonal part of the stress tensor. To ensure the effect of strong correlations within this limit, it is necessary to force the hierarchy t < ω ≪ U . We enable this by projecting out double occupancy using the Gutzwiller projector, hence the t-J model is the appropriate description. Our plan is as follows: we show the connection of this high-frequency limit - not measurable in experiments to the dc or low-frequency behavior. Then we evaluate the dependence on the parameters T , J and x for several ∗ systems. Hence, for the first part we compare RH with the expression derived from the Kubo formulae for the electrical conductivities
1 X exp −(βǫν ) − exp −(βǫµ ) >− < ν|Jα |µ >< µ|Jβ |ν > Z µ,ν ǫµ − ǫν − ω − iη
ˆ and Z the canonwith ǫν the ν’th energy eigenvalue of H ical partition function. In terms of this expression the Hall coefficient is given by σx,y (ω) ∂ . (5) RH (ω) = v ∂B σx,x (ω)σy,y (ω) B=0 As a transport object RH is a demanding quantity to study on finite clusters. The effect of boundary conditions and their finite size implications are difficult to eliminate. We introduce a small magnetic field through the Peierls substitution (Eq. 2) which impacts only on the orbital motion of the electrons and does not couple
(3)
!
,
(4)
to the spin. Studying RH on toroidal and spherical geometries reduces the finite size effects on the transport, however, these periodic systems constrain the smallest accessible flux φ0 to π/L. For L = 12 this is already a considerable flux. An alternative are ladder systems such as in Fig. 1(b) where the limit φ0 → 0 is possible. However, ladders have additional finite size effects impacting on the transport quantities. For several dopings (x = 0.08, 0.67, 0.75, 0.83) we compute RH (ω) as a function of temperature both on the torus and the icosahedron. To take into account the finite size of the system, a broadening η of the energy equal
10
2
hEi0 , hEi0 the g.s. energy D the charge stiffness D = d dφ 2 and φ a constant flux inducing a persistent current41 . In this fashion, we evaluated the doping dependence of the 0 g.s. and found RH > 0 in the experimental regime near 0 x ≈ 0.7 and a divergence RH → −∞ as the Mott-limit 11 is approached . The second check refers back to the basic definition of the dc-Hall-coefficient RH = −Ey /(Bjx ) and computing this directly on a ladder system. This computation showed no change of sign for x ≈ 0.7 as function of T and a qualitatively similar behavior to the ω = ∞ result.
14 12
ω=0
ω=3|t|
ω=∞
10 x = 0.08
e0RH(T)/v
8 0.83
6 0.75
4
0.67
2 0 -2 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 T/|t| T/|t| T/|t|
FIG. 9: (color on-line) RH (ω, T ) on 12-site torus at several frequencies for several dopings and J = 0.
16 14 12 10 8 6 4 2 e0RH(ω)/v
to the mean level spacing of the current matrix elements was employed, this requires η ≈ 3|t|. In Fig. 9 we show RH as function of temperature at three different frequencies. Comparing ω = 0 to ω = ∞ we find that the dclimit is qualitatively the same as the high-frequency limit, however, the overall temperature-dependence is scaled by a renormalization factor s ≡ RH (T, ω = 0)/RH (T, ∞) where s is only weakly temperature dependent. For the torus, s was found to be 0.83 (0.78) at x = 0.83 (x = 0.75). At x = 0.083 the many-body renormalization is stronger. Conversely, in Fig. 10 we show the frequencydependence of RH at several temperatures and different dopings. Similar to the work done by Grayson et al.40 on the Hall angle, finite frequency experiments may be helpful to confirm the weak frequency-dependence seen in our computations: With |t| ≈ 10meV results in the frequency range ω ≥ 3|t| ≈ 30meV could be compared to our theoretical results and hence lend experimental support for the usefulness of the high-frequency limit3 of RH . We have computed two additional quantities related to the dc-limit as checks of the validity of the dccomputations. Zotos et al.29 derived an expression for the 1 d g.s. expectation value, RH (T = 0) = |e| dn log D(n) with
4
x=0.08
2 1 T/|t| = .5
6 5 4 3 2 8 7 6 5 4 3
x=0.67
x=0.75
0
1
2
3 ω/|t|
4
5
6
FIG. 10: (color on-line) RH (ω) at several temperatures and dopings as function of frequency on 12-site Torus a for J = 0. Temperatures in all plots T /|t| = (0.5, 1, 2, 4).
Encouraged by the notion of weak frequency∗ dependence, we studied RH ≡ RH (ω = ∞). We have studied several 12 site tori, several doping values on an 18 site torus, the icosahedron and several ladders. ∗ The g.s. RH (x) shows a Drude drop-off near the band insulator and a saddle point around x = 0.5 as was reported by the authors in ref. 11. When the strongcorrelation regime near the Mott-insulator is approached (x < 1/2), the slope again increases and for 1 hole 0 (x = 0.083) RH changes sign. The high temperature behavior confirms the prediction from the high-temperature series expansion3,12 . The interesting temperature range is for T < 2|t| where the experimental data shows an oscillatory behavior and analytical results are not available (Fig. 11). Our computations show a distinct minimum in RH (T ) which appears near T = |t| in the Curie-Weiss regime (x ≈ 0.7) and continuously moves to smaller temperatures as the doping is decreased. For the smallest dopings accessible (x = 1/L) this minimum appears near T = 0.2|t|, while the behavior becomes linear at higher T in all cases. In the Curie-Weiss regime some systems also show a weak
E.
Diamagnetic Susceptibility
Lastly, we study the orbital susceptibility per site χd (T ) ≡ −
1 ∂2F , L ∂B 2
where F is the free energy and B the magnitude of the perpendicular magnetic field. We define the dimensionless flux α = BA/φ0 through a triangular plaquette with
10 9 8 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 7 6
x=0.08
.6 J/|t|
x=0.16
0
J/|t| x=0.08 0.6
0
0.3
x=0.67
x=0.16 0.6
0.2
.6 J/|t|
0.1 0
0 0.3
0.6
0.2
0
0
5
.6
4 3 8 7 6 5 4 3
0.4 0.3 0.2 0.1 0 0.4
0
cv(T)/kB
maximum close to T = 0.25|t|. These results point towards the existence of two distinct energy scales. The pronounced minimum appears to be correlated with the peak in the electronic specific heat (Fig. 11), which also occurs around T = |t| in the band limit, and gradually shifts towards smaller temperatures with decreased doping. To strengthen this point it is useful to include the dependence on J. As reported for the specific heat, the effect of J is to move the peak in specific heat to larger temperatures. This behavior is mirrored by RH . In Fig. 11 we show RH (T ) for several dopings and J. At x = 0.75 the peak of Cv (T ) shifts from T = 1.3|t| to T = 1.5|t| while the minimum in RH (T ) moves from T = 0.75|t| to T = 1.1|t|. At x = 0.67 the behavior around T = |t| is similar, however an additional peak in the specific heat emerges at T < 0.5|t|. A qualitatively similar feature is seen in other clusters, however, its magnitude is dependent on the system geometry. Hence, we do not attempt to extract quantitative information in the T -range below T = 0.5|t|. Nonetheless, it is important to note that with decreasing doping a second energy scale - caused by spin interactions - becomes dominant. At small doping x = 0.16 and x = 0.08 the specific heat becomes strongly J-dependent and the main peak in the specific heat as discussed in the first section - increases significantly with J. The variations in RH (T ) with J are also more pronounced. Assuming t ≈ −100 K we compare our results to the experimentally available data6 at x = 0.31 and x = 0.71, this was done in ref. 11 for x = 0.71. In Fig. 12 the comparison at x = 0.31 shows good agreement for the slope at high T and reproduces the temperature scale for the minimum near 25 K. For x = 0.71 the temperature scale for the minimum near 100 K is seen both in experiment and theory. The slope of the high temperature limit differs by a factor of 4 and was adjusted in the plot for better comparison. Also, the theory does not show the change of sign as function of temperature. However, the oscillatory behavior below T = 100 K and the onset of the linear increase for T > 200 K agrees well with experiment. Hence, the exact temperature dependence of RH remains a challenging task - both on the theoretical and experimental side.
e0RH*(T)/v
11
J/|t|
x=0.75
x=0.67
0.1 0 0.3
0.6 0
0.2
0 .6 J/|t| 0 0.5 1 1.5 2 2.5 3 T/|t|
0.1
x=0.75
0 0.5 1 1.5 2 2.5 3 T/|t|
∗ FIG. 11: (color on-line) (left) RH (T ) for several x and J for Torus b. (right) Cv (T ) for several x and J on same Torus.
√ A = 3a2 /4 the area of the plaquette and a the lattice spacing. We get 2 1 ∂ 2 F ∂α . χd = − L ∂α2 ∂B It is useful to employ ladder systems as shown in Fig. 1(b). Defining t = ~2 /2ml2 we have 1 1 ∂f 2m χd = − 2 2 A2 µ2B , L ~ l L ∂α where f is the dimensionless F/t and µB is the Bohrmagneton. In Fig. 13(a) we present the diamagnetic susceptibility and its first two derivatives with respect to T . For free electrons in an infinite system, χd corresponds to electrons moving in circular orbits with radii inversely proportional to the magnetic field. RH measures the polarization induced in a system with a stationary current and perpendicular magnetic field. It is hence reasonable to ask for a connection between these two quantities. Our numerics suggest the relation T
∂ 2 χd ∂ 2 RH = c(x) 2 dT ∂T 2
(6)
where c(x) is a function depending on doping x. In Fig. 13(b) we display our numerical result ∗ for RH (T ) at a doping relevant to the experimental situation6 compared to the result obtained from integrating Eqtn. (6) and choosing the two integration constants
20
1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 9 8 7 6 5 4 3 2 1 0 -1
x=0.31, Exp.
15 2
arb. units
(a)
2
χd=-d F/dB dχ /dT 2 d 2 d χd/dT drH/dT rH
10 5 0
x=0.25 (Torus)
-5
x=0.67 (Torus)
-10 0
0.5
1
1.5
2 2.5 T/|t|
3
3.5
4
3.5
4
7.5
x=0.67 (Icos.)
R H* from χd
7
x=0.71, Exp.
6.5
0
100
200 T/K
300
400
FIG. 12: (color on-line) Comparison of 12-site Torus b and icosahedron with experiment36 at x = 0.31 and x = 0.71, we take t = −100 K, in the lower plot our numerical results are scaled to match the slope of the linear part at T > 200 K in the experimental results.
appropriately. For J = 0 this was done by the authors in ref. 11. Ignoring the range for T < 0.5|t| which may be strongly influenced by finite size effects, we find that both curves show a minimum at T ≈ |t| and then increase until T = 2|t|. The reaction to finite J > 0 is also similar for both curves, moving the minimum from T = |t| to roughly T = 1.25|t|. We plan to further investigate this connection within a high-temperature expansion for χd (T ), in particular for the doping regime close to the Mott-insulator.
IV.
CONCLUSION
In summary, we have presented a systematic study of the entire doping regime for the triangular lattice t-J model with electronic frustration (t < 0 for n > 1). We have identified low energy excitations, stemming purely from kinetic processes (J = 0) of infinitely correlated spin- 21 fermions. These are manifest in a peak structure in the specific heat emerging at very low T near the Mottinsulator, however persisting for intermediate dopings. The effect of electronic frustration is further supported by entropy-enhancement in the underdoped regime with a low entropy recovery temperature.
(b)
e0RH(T)/v
RH (10-9m3/C)
12
6
J/|t|=0
5.5
.4
5 4.5 4 3.5 0
0.5
1
1.5
2 2.5 T/|t|
3
FIG. 13: (color on-line) (a) χd (T ) and unitless Hall-coefficient rH and their derivatives on a 12-site ladder at x = 0.83. (b) ∗ Comparison of numerical results for x = 0.83 for RH (T ) (obtained on an 18-site torus) with the integral of Eqtn. (6) (χd (T ) obtained on a 12-site ladder system) for two different values of J. Appeared partly in ref. 11.
We have studied both the diamagnetic and spinsusceptibility. For the spin susceptibility we extract a relation for effective spin-interactions θ(x, J) valid for all dopings. This stands in contrast to Anderson’s formula for FM t > 0, since in our case AFM is supported rather than suppressed by doping. In the experimentally relevant doping range of x > 0.5 we offer an explanation for the crossover between the Curie-Weiss and SDW metallic phases, as observed in experiments7 . For the diamagnetic susceptibility χd a connection is obtained between χd and the temperature dependent Hall coefficient. The dynamic Hall coefficient reveals a weak frequencydependence and a qualitatively similar dc and highfrequency behavior, with stronger renormalizations close to the Mott-limit. This strengthens the practical ∗ relevance of the high-frequency limit3 RH . The temperature-dependence confirms the linear slope at high temperatures3 and reproduces the experimentally ob-
13 served minimum near T = 100 K at x = 0.71. In conclusion, the t-J model is applicable to NCO for x > 0.5 and shows that the interface of strong correlations and metallic behavior indeed describes the experimental situation here. For x < 0.5 the frustrated t-J model leads to subtle effects of strong correlations, even in the case of J = 0. Our results show, that for the proper description of experimentally observed normal state properties7 there - such as the charge-ordered insu-
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Acknowledgments
This work is supported by Grant No. nsf-dmr0408247.
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